cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A264413 G.f. A(x) satisfies: A(x)^2 = A(x^2) + 12*x.

Original entry on oeis.org

1, 6, -15, 90, -660, 5310, -45765, 413640, -3864345, 37014120, -361577790, 3588484140, -36079979085, 366728363460, -3762120325140, 38901621985290, -405039437707575, 4242802537386450, -44681704461745740, 472795814216587140, -5024232597805717410, 53596341229925979360, -573736849659978481665, 6161218734911098973490, -66355728143871653462745
Offset: 0

Views

Author

Paul D. Hanna, Nov 12 2015

Keywords

Examples

			G.f.: A(x) = 1 + 6*x - 15*x^2 + 90*x^3 - 660*x^4 + 5310*x^5 - 45765*x^6 + 413640*x^7 - 3864345*x^8 + 37014120*x^9 - 361577790*x^10 +...
where
A(x)^2 = 1 + 12*x + 6*x^2 - 15*x^4 + 90*x^6 - 660*x^8 + 5310*x^10 - 45765*x^12 + 413640*x^14 - 3864345*x^16 + 37014120*x^18 - 361577790*x^20 +...
so that A(x)^2 = A(x^2) + 12*x.
Let G(x) = Series_Reversion( x / (A(x^2) + 3*x) ), then
G(x) = x + 3*x^2 + 15*x^3 + 81*x^4 + 462*x^5 + 2718*x^6 + 16344*x^7 + 99900*x^8 + 618567*x^9 + 3870909*x^10 +...+ A264225(n)*x^n +...
such that G(x)^2 = G( x^2/(1-6*x) ).

Links

Crossrefs

Cf. A271935, A264225, A264233, A264412, A264414, A264415.

Programs

  • PARI
    {a(n) = my(A=1); for(i=1,n, A = sqrt( subst(A,x,x^2) + 12*x +x*O(x^n))); polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))

Formula

Given g.f. A(x), let G(x) denote the g.f. of A264225, then:
(1) G( x/(A(x)^2 - 9*x) ) = x,
(2) G( x/(A(x^2) + 3*x) ) = x,
(3) A(G(x))^2 = (1+9*x) * G(x)/x,
(4) A(G(x)^2) = (1-3*x) * G(x)/x,
where G(x)^2 = G( x^2/(1-6*x) ).

A271930 G.f. A(x) satisfies: A(x) = A( x^2 + 6*x*A(x)^2 )^(1/2), with A(0)=0, A'(0)=1.

Original entry on oeis.org

1, 3, 15, 90, 597, 4221, 31185, 237897, 1859568, 14816637, 119892942, 982565883, 8138777166, 68028775587, 573078135996, 4860507197700, 41470162208814, 355695498901179, 3065210379987489, 26525947283576640, 230425563258798840, 2008561878414115803, 17563090615911038115, 154014411705019299450, 1354142406561753259035, 11934928413519024726252, 105426063390991627937457, 933206579920813459523994, 8276480132736299734057275, 73535083052134446419214960
Offset: 1

Views

Author

Paul D. Hanna, Apr 16 2016

Keywords

Comments

Compare the g.f. to the following related identities:
(1) C(x) = C( x^2 + 2*x*C(x)^2 )^(1/2), where C(x) = x + C(x)^2 (A000108).
(2) F(x) = F( x^2 + 4*x*F(x)^2 )^(1/2), where F(x) = D(x)^2/x and D(x) = x + D(x)^3/x (A001764).

Examples

			G..f.: A(x) = x + 3*x^2 + 15*x^3 + 90*x^4 + 597*x^5 + 4221*x^6 + 31185*x^7 + 237897*x^8 + 1859568*x^9 + 14816637*x^10 + 119892942*x^11 + 982565883*x^12 +...
where A(x)^2 = A( x^2 + 6*x*A(x)^2 ).
RELATED SERIES.
A(x)^2 = x^2 + 6*x^3 + 39*x^4 + 270*x^5 + 1959*x^6 + 14724*x^7 + 113706*x^8 + 896994*x^9 + 7198257*x^10 + 58580766*x^11 + 482345937*x^12 + 4011023556*x^13 + 33637887441*x^14 +...
Let B(x) be the series reversion of the g.f. A(x), A(B(x)) = x, then:
B(x) = x - 3*x^2 + 3*x^3 - 3*x^5 + 9*x^7 - 33*x^9 + 126*x^11 - 513*x^13 + 2214*x^15 - 9876*x^17 + 45045*x^19 - 209493*x^21 +...+ A264412(n)*x^(2*n+1) +...
such that B(x) = x*G(x^2) - 3*x^2 where G(x)^2 = G(x^2) + 6*x, and G(x) is the g.f. of A264412.

Links

Crossrefs

Cf. A264412, A271935, A271957, A271931, A271934.

Programs

  • PARI
    {a(n) = my(A=x,X=x+x*O(x^n)); for(i=1,n, A = subst(A,x, x^2 + 6*X*A^2)^(1/2) ); polcoeff(A,n)}
for(n=1,30,print1(a(n),", "))

Formula

G.f. A(x) satisfies: A( x*G(x^2) - 3*x^2 ) = x, where G(x)^2 = G(x^2) + 6*x, and G(x) is the g.f. of A264412.
a(n) ~ c * d^n / n^(3/2), where d = 9.35010183959428615991060685319... and c = 0.0902227396498060205291555743... . - Vaclav Kotesovec, Apr 18 2016

A271957 G.f. A(x) satisfies: A(x) = A( x^2 + 10*x*A(x)^2 )^(1/2), with A(0)=0, A'(0)=1.

Original entry on oeis.org

1, 5, 40, 375, 3845, 41825, 474450, 5552250, 66548785, 812875800, 10082125950, 126637168125, 1607562407775, 20591392666250, 265810034489750, 3454516382881875, 45162288467005155, 593528625987396725, 7836767285955169200, 103908861022437312375, 1382961699685548183750, 18469547560714428659250, 247433242662040209056250, 3324296142183357299203125, 44779542961314348791789400, 604655933814703316140014375
Offset: 1

Views

Author

Paul D. Hanna, Apr 17 2016

Keywords

Comments

Compare the g.f. to the following related identities:
(1) C(x) = C( x^2 + 2*x*C(x)^2 )^(1/2), where C(x) = x + C(x)^2 (A000108).
(2) F(x) = F( x^2 + 4*x*F(x)^2 )^(1/2), where F(x) = D(x)^2/x and D(x) = x + D(x)^3/x (A001764).

Examples

			G..f.: A(x) = x + 5*x^2 + 40*x^3 + 375*x^4 + 3845*x^5 + 41825*x^6 + 474450*x^7 + 5552250*x^8 + 66548785*x^9 + 812875800*x^10 + 10082125950*x^11 + 126637168125*x^12 +...
where A(x)^2 = A( x^2 + 10*x*A(x)^2 ).
RELATED SERIES.
A(x)^2 = x^2 + 10*x^3 + 105*x^4 + 1150*x^5 + 13040*x^6 + 152100*x^7 + 1815375*x^8 + 22078750*x^9 + 272728845*x^10 + 3412891200*x^11 + 43178951325*x^12 +...
Let B(x) be the series reversion of the g.f. A(x), A(B(x)) = x, then:
B(x) = x - 5*x^2 + 10*x^3 - 45*x^5 + 450*x^7 - 5535*x^9 + 75600*x^11 - 1106100*x^13 + 16953750*x^15 +...+ A264414(n)*x^(2*n+1) +...
such that B(x) = x*G(x^2) - 5*x^2 where G(x)^2 = G(x^2) + 20*x, and G(x) is the g.f. of A264414.

Links

Crossrefs

Cf. A264414, A271930, A271935, A271931, A271934.

Programs

  • PARI
    {a(n) = my(A=x+x^2,X=x+x*O(x^n)); for(i=1,n, A = subst(A,x, x^2 + 10*X*A^2)^(1/2) ); polcoeff(A,n)}
for(n=1,30,print1(a(n),", "))

Formula

G.f. A(x) satisfies: A( x*G(x^2) - 5*x^2 ) = x, where G(x)^2 = G(x^2) + 20*x, and G(x) is the g.f. of A264414.
Showing 1-3 of 3 results.

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